# Prove that the graph of a measurable function is measurable

1. May 28, 2012

### sunjin09

1. The problem statement, all variables and given/known data
Let f: X->R be measurable, prove that Z={(x,y)|y=f(x)} is a measurable set of XxR.

2. Relevant equations

A subset Z of XxR is measurable iff Z is a countable union of product of measurable sets of X and R.

3. The attempt at a solution

Let $R=\cup_kV_k$, where $V_k$ are finite intervals of R, let $U_k=f^{-1}(V_k)$, obviously $Z\subset\cup_kU_k\times V_k$ where $\cup_kU_k\times V_k$ is measurable since both $U_k$ and $V_k$ are measurable. How to I proceed to show that Z as a subset of a measurable set of XxR is also measurable?